Proof of Proposition REF :
For the random variable \(\eta \) , using (REF ), we can show that (see Chapter 8 in [1]})
\({h}(\eta )&=~ \dfrac{1}{2}\log \left( \mathrm {det} (\sigma ^2 \mathbf {S}^{-1}) \right)+\dfrac{n}{2}\log (2\pi e) \nonumber \\ &= ~\dfrac{1}{2} \sum _{i=1}^n \log \left(\dfrac{\sigma ^2}{\lambda _i(\mathbf {S})}\right) +\dfrac{n}{2}\log (2\pi e) \\&= \frac{1}{2} \rho _e(\Phi ) + \frac{n}{2} \big (1+ \log (2 \pi \sigma ^2) \big ).\) 
